Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms, 2nd Edition

1Basics of Fourier Analysis

1.1 Forward and Inverse Fourier Transform

Fourier transform (FT) is a common and useful mathematical tool that is utilized in innumerous applications in science and technology. FT is quite practical especially for characterizing nonlinear functions in nonlinear systems, analyzing random signals, and solving linear problems. FT is also a very important tool in radar imaging applications as we shall investigate in the forthcoming chapters of this book. Before starting to deal with the FT and inverse Fourier transform (IFT), a brief history of this useful linear operator, and its founders are presented.

1.1.1 Brief History of FT

Jean Baptiste Joseph Fourier, a great mathematician, was born in 1768, Auxerre, France. His special interest in heat conduction led him to describe a mathematical series of sine and cosine terms that could be used to analyze propagation and diffusion of heat in solid bodies. In 1807, he tried to share his innovative ideas with researchers by preparing an essay entitled as On the Propagation of Heat in Solid Bodies. The work was examined by Lagrange, Laplace, Monge, and Lacroix. Lagrange's oppositions caused the rejection of Fourier's paper. This unfortunate decision cost colleagues to wait for 15 more years to meet his remarkable contributions to mathematics, physics, and especially on signal analysis. Finally, his ideas were published thru the book The Analytic Theory of Heat in 1822 (Fourier 1955).

Discrete Fourier transform ...